Show ring has no divisors of zero

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In summary, the conversation discusses the problem of proving that a ring containing two or more elements, in which every nonzero element has a unique inverse under a special operation, does not contain any divisors of zero. One participant begins by assuming that such a ring does contain divisors of zero, and proceeds to show that this leads to a contradiction. The other participant offers a hint to consider the expression a(b+c)a and suggests that the conversation is at a standstill in terms of progress.
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samkolb
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Homework Statement


Let R be a ring containing two or more elements such that for each nonzero a in R, there exists a unique b in R such that a=aba. Show that R contains no divisors of zero.


Homework Equations


The ring axioms.



The Attempt at a Solution


I assumed that R contains divisors of zero. So assume there exist nonzero a,b in R such that ab=0. Then there exist unique c,d in R such that a=aca and b=bdb.

Consider the expression ca+bd.

If ca+bd=0, then multiplying on the left by a gives a=0 and multiplying on the right by b gives b=0.

So assume ca+bd is not equal to zero. Then there exists unique e in R such that
ca+bd=(ca+bd)e(ca+bd).

Multiplying on the left by a gives a=aebd+aeca=ae(ca+bd).

Substituting this expression for a into a=aca gives
a=aca=ae(ca+bd)ca.

From the uniqueness of c, e(ca+bd)c=c.

This is as far as I've gotten. I don't know if I have made any progress or am just going in circles.
 
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  • #2
Suppose a and c are zero divisors. Then ac=0. There is a UNIQUE b such that aba=a. What is a(b+c)a?
 

What does it mean for a show ring to have no divisors of zero?

When we say that a show ring has no divisors of zero, it means that there are no elements in the ring that when multiplied by any other element in the ring, will result in a product of zero.

Why is it important for a show ring to have no divisors of zero?

A show ring with no divisors of zero is important because it ensures that the ring follows the fundamental property of rings, which is closure under multiplication. This means that every pair of elements in the ring will produce another element in the ring when multiplied, and there will be no exceptions.

What are the consequences of a show ring having divisors of zero?

If a show ring has divisors of zero, it violates the closure property and leads to inconsistencies and contradictions in mathematical operations. This can make it difficult to perform calculations and make deductions based on the properties of the ring.

Can a show ring have more than one divisor of zero?

No, a show ring cannot have more than one divisor of zero. In fact, if a show ring has even one divisor of zero, it automatically means that every element in the ring is a divisor of zero. This is because if a x b = 0, then either a or b must be 0, making it a divisor of zero.

How can we determine if a show ring has divisors of zero?

To determine if a show ring has divisors of zero, we can use the zero divisor property. This states that if a x b = 0, then either a or b (or both) must be a divisor of zero. By checking all pairs of elements in the ring and seeing if any of them result in a product of 0, we can determine if the ring has divisors of zero.

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